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12. The development of

(Burton , 8.3 – 8.4 )

We have already noted that certain ancient Greek mathematicians — most notably Eudoxus and — had successfully studied some of the basic problems and ideas from calculus , and we noted that their was similar to the modern approach in some respects (successive geometric approximations using figures whose measurements were already known) and different in others (there was no concept , and instead there were delicate reductio ad absurdum arguments) . Towards the end of the 16 th century there was further interest in the sorts of problems that Archimedes studied , and later in the 17 th century this led to the independent development of calculus in Europe by (1643 – 1727) and Gottfried Wilhelm von Leibniz (1646 – 1716) .

Two important factors led to the development of calculus originated in the late Middle Ages , and both represented attempts to move beyond the bounds of ancient Greek mathematics , philosophy and physics .

1. Interest in questions about infinite processes and objects . As noted earlier , ancient Greek mathematics was not equipped to deal effectively with Zeno ’s paradoxes about infinite and by the time of Aristotle it had essentially insulated itself from its “horror of the infinite .” However , we have also noted that Indian mathematicians did not share this reluctance to work with the concept of infinity , and Chinese mathematicians also used methods like infinite when these were convenient . During the 14 th century the mathematical work of Oresme and others on infinite series complemented the interests of the scholastic philosophers ; questions about the infinite played a key role in the efforts of scholastic philosophy to integrate Greek philosophy with Christianity . Philosophical writings by scholars including William of Ockham (1285 – 1349) , Gregory of Rimini (1300 – 1358) , and Nicholas of Cusa (1401 – 1464) provided insights which anticipated the concept of limit — a basic part of calculus as we know it .

2. Interest in questions about physical motion . During the 13 th century Jordanus de Nemore discovered the mathematically correct description for the physics of an inclined plane (a result that had eluded Archimedes and was described incorrectly by Pappus) , and later members of the Merton school in Oxford such as T. Bradwardine (1290 – 1349) and W. Heytesbury (1313 – 1373) had discovered an important property of uniformly accelerated motion ; namely , the average velocity (total divided by total time) is in fact the mathematical average of the initial and final velocities . The significance of this concept for the motion of freely falling bodies was not understood at the time and would not be known until the work of Galileo and others in the 16 th century . During the late 16 th century interaction between mathematics and physics began to increase at a much faster rate , and many important contributors to mathematics such as S. Stevin also made important contributions to physics (in Stevin ’s case , the centers of mass for certain objects and the statics and dynamics of fluids) .

Here are some online references to further information about the topics from the Middle Ages described above :

http://www.math.tamu.edu/%7Edallen/masters/medieval/medieval.pdf

http://www.math.tamu.edu/%7Edallen/masters/infinity/infinity.pdf

http://plato.stanford.edu/entries/heytesbury/

Problems leading to the development of calculus

In the early 17 th century mathematicians became interested in several types of problems , partly because these were motivated by advances in physics and partly because they were viewed as interesting in their own right .

1. Measurement questions such as of , areas of planar regions and surfaces , and volumes of solid regions , and also finding the centers of mass for such objects .

2. Geometric and physical attributes of curves , such as and normals to curves , the concept of , and the relation of these to questions about velocity and acceleration .

3. Maximum and minimum principles ; e.g ., the maximum height achieved by a projectile in motion or the greatest area enclosed by a rectangle with a given .

Specific examples of all three types of problems had already been studied by Greek mathematicians . The results of Eudoxus and Archimedes on the first type or problem and of Archimedes and Apollonius on the second have already been discussed . We know about the work of Zenodorus (200 – 140 B.C.E.) on problems in the third type because it is reproduced in Pappus ’ anthology of mathematical works (the previously cited Collection or Synagoge ). Some examples of Zenodurus ’ results are

(1) among all regular n – gons with a fixed perimeter , a regular n – gon encloses the greatest area , (2) given a and a such that the perimeter of the polygon equals the of the circle , the latter encloses the greater area , (3) a encloses the greatest volume of all surfaces with a fixed .

GIVE REFERENCES

Infinitesimals and Cavalieri ’s Principle

We shall attempt to illustrate the concept of with an example of its use to determine the volume of a geometrical figure .

Usually the formula for the volume of a cone is derived only for a right circular cone in which the axis through the vertex or nappe is perpendicular to the plane of the base. It is naturally to ask whether there is a similar formula for the volume of an oblique cone for which the axis line is not perpendicular to the plane of the base (see the drawing below). One can answer this question using an approach developed by B. Cavalieri (1598 – 1647) . Similar ideas had been considered by Zu Chongzhi , also known as Tsu Ch ’ung – chin (429 – 501) , but the latter ’s work was not known to Europeans at the time .

CAVALIERI’S PRINCIPLE. Suppose that we have a pair three-dimensional solids S and T that lie between two parallel planes P1 and P2, and suppose further that for each plane Q that is parallel to the latter and between them the plane sections Q ∩∩∩ S and Q ∩∩∩ T have equal areas . Then the volumes of S and T are equal .

Here is a physical demonstration which suggests this result : Take two identical decks of cards that are neatly stacked , just as they come right out of the package . Leave one untouched , and for the second deck push along one of the vertical edges so that the deck forms a rectangular parallelepiped as below .

In this new configuration the second deck has the same volume as first and it is built out of very thin rectangular pieces (the individual cards) whose areas are the same as those of the corresponding cards in the first deck . So the areas of the plane sections given by the separate cards are the same and the volumes of the solids formed from the decks are also equal .

We shall now apply this principle to cones . Suppose we have an oblique cone as on the right hand side of the figure below . On the left hand side suppose we have a right circular cone with the same height and a circular base whose area is equal to that of the elliptical base for the second cone .

Figure illustrating Cavalieri’s Principle

In the notation of Cavalieri ’s Principle , we can take P1 to be the plane containing the bases of the two cones and P2 to be the plane which contains their vertices and is parallel to P1. Let Q be a plane that is parallel to both of them such that the constant distance between P2 and Q is equal to y; we shall let h denote the distance between P1 and P2, so that h is also the altitude of both cones . If b denotes the areas of the bases of both cones then the areas of the sections formed by intersecting these cones with Q 2 2 are both equal to b y / h . Therefore Cavalieri ’s Principle implies that both cones have 2 the same volume , and since the volume of the right circular cone is b h / 3 it follows that the same is true for the oblique cone .

One approach to phrasing our physical motivation in mathematical terms is to imagine each cone as being a union of a family of solid regions given by the plane sections ; for the right circular cone these are regions bounded by , and for the oblique cone they are bounded by . Suppose we think of these sections as representing cylinders that are extremely thin . Then in each case one can imagine that the volume is formed by adding together the volumes of these cylinders , whose areas — and presumably thicknesses — are all the same , and of course this implies that the volumes of the original figures are the same . One fundamental question in this approach is to be more specific about the meaning of “extremely thin .” Since a planar figure has no finite thickness , one might imagine that the thickness is something less , and this is how one is led to the concept of a thickness that is infinitesimally small.

The preceding discussion suggests that an quantity is supposed to be nonzero but is in some sense smaller than any finite quantity . If we are given a geometric figure that can be viewed as a union of “indivisible ” objects with one less dimension — for example , the planar region bounded by a rectangle viewed as a union of line segments parallel to two of the sides , or the solid region bounded by a cube as a union of planar regions bounded by squares parallel to two of the faces — then the idea is to view the square as a union of rectangular regions with infinitesimally small width or the square as a union of solid rectangular regions with infinitesimally small height. Likewise , this approach suggests an interpretation of a continuous as being composed of a family of straight lines with infinitesimally small . When applied to our examples of cones , it leads to thinking of the solid region bounded by either cone as a union of circular or elliptical cylinders with infinitesimally small height .

Mathematicians and users of mathematics have though about infinitesimals for a long time . They already appear in the mathematics of the early Greek atomist philosopher Democritus (460 – 370 B.C.E.) and an approach to squaring the circle developed by Antiphon , but advances by Eudoxus and others during the 4 th century B.C.E. enabled Greek mathematicians to avoid the concept , and this fit perfectly with the reluctance of Greek mathematicians and philosophers ( e.g ., Aristotle) to eliminate questions about the infinite from their mathematics . Taking the somewhat obscure form of “indivisibles ,” they reappeared in the mathematics of the late Middle Ages , and they played an important role in the work of J. Kepler (1571 – 1630) on laws of planetary motion , particularly his Second Law which states that the orbits of planets around the sun sweep out equal areas over equal times . During the 17 th century infinitesimals were used freely by many mathematicians and scientists who contributed to the development of calculus , and in particular both Newton and Leibniz used the concept in their definitive accounts of the subject . However , as calculus continued to develop , doubts about the logical soundness of infinitesimals also began to mount . Such questions ultimately had very important consequences for the development of mathematics, and they will be discussed later in the next unit .

A brief discussion of Cavalieri ’s Theorem , its importance for classical geometry , and its interpretation in terms of integral calculus appears on pages 156 – 159 (= document pages 15 – 17) of the following online reference :

http://math.ucr.edu/~res/math133/geometrynotes3c.pdf

A general comment on coverage and attribution

During the 17 th century many mathematicians were interested in similar problems , and many results were discovered independently by two or more researchers . Not all such cases can be described completely in a brief summary such as these notes ; one guiding principle here is to mention the persons whose work on a given problem had the most impact . EVES QUOTE

Progress on measurement questions

Methods and results from Archimedes and others provided important background and motivation for work in this area . We have already discussed the use of infinitesimals to derive formulas in some cases , and there was a great deal of further work based upon such ideas . In particular , during the time before the appearance of Newton ’s and Leibniz ’ work , many of the standard examples in integral calculus had been worked out by preliminary versions of methods that became standard parts of the subject . Here are some specific examples :

Integrals of polynomials and more general power functions. Cavalieri computed n the integral of x geometrically in cases where n is a positive integer , Gregory of St. Vincent (1584 – 1667) integrated 1/x in geometric terms that are equivalent to the usual formula involving log e x, and J. Wallis (1616 – 1703) generalized the integral formula for n x to other real values of n. — Wallis was a particularly important figure in the development of calculus for several reasons . His methods , which are discussed on pages 357 – 360 of Burton , replaced geometric techniques with algebraic computations and analytic considerations , and as such they are a milestone in the development of analysis (calculus) as a subject distinct from both algebra and geometry . In an entirely different direction , Wallis is also known today for his applications of integral formulas to derive his infinite product formula for πππ/2:

We should note here that an earlier infinite product formula involving πππ

had been discovered by Viète (and , as noted earlier , Indian mathematicians had already discovered some of the standard infinite series formulas involving πππ). REFERENCES osler

Measurements involving the curve and regions partially bounded by it. There was an enormous amount of interest in the properties of the cycloid curve during the 17 th century , and there were also some bitter disputes about priorities among some of the numerous mathematicians who worked on this example . Results included computations of arc lengths as well as areas , volumes and centers of mass associated to the curve . Mathematicians whose names are associated with this work include B. Pascal (1623 – 1662) , E. Torricelli (1608 – 1647) , and G. P. de Roberval (1602 – 1675) .

Infinite solids with finite volume. Torricelli also discovered a fact that few if any mathematicians had anticipated ; namely , the existence of an unbounded solid of revolution whose volume is finite . His example is an unbounded piece of the solid

formed by rotating the standard y = 1/ x about the x – axis .

Arc length. Early in the 17th century there were doubts about the possibility of computing the arc lengths of many curves , including some extremely familiar examples . Results of H. van Heuraet (1633 – 1660) showed the problem of finding arc length of a given curve is equivalent to determining the area under another curve , and he also 2 3 worked out certain examples including the semi – cubical a y = x . The arc length of a spiral curve was computed by Roberval .

Integrals and series expansions of transcendental functions. Results on the integrals of the standard trigonometric functions were obtained by Pascal , Roberval , I. Barrow (1630 – 1677) and J. Gregory (1638 – 1675) . The standard infinite series expansion for arctan x was obtained by Gregory (however , as noted before , the Indian mathematician Madhava had discovered the formula two centuries earlier) , and the

standard infinite series expansion for log e (1 + x) was obtained by N. Mercator (1620 – 1687) ; the latter should not be confused with the mathematical cartographer G. Mercator mentioned earlier . Gregory also made numerous other contributions , including extending and applying the classical method of exhaustion to questions about other conic sections and writing the first text covering the material that would become calculus .

Progress in differentiation and maximization /minimization problems

Greek mathematics provided far less insight into questions about lines or maximizing functions than it did for computing areas and volumes . In particular , there were no general principles comparable to the method of exhaustion for describing tangents ; each example was treated in an entirely separate manner . Therefore one major problem facing 17 th century mathematicians was to produce a workable concept of tangent line .

Some names particularly associated with this problem are Descartes , Fermat , Roberval and Barrow . Descartes ’ approach was based on finding the normal (perpendicular) line to a curve at a , and Roberval ’s was motivated by the standard interpretation of a parametrized curve as the path of a moving object . Fermat and Barrow both defined tangent lines by the method that has become standard ; namely , the tangent line to a curve C at a point x is a limit of the secant lines joining x to a second point y on C as y approaches x. Fermat understood the basic idea , but Barrow’s language ( i.e ., his differential triangle ) was more precise . A discussion of Barrow ’s work appears on pages 363 – 364 of Burton .

As in the case of measurement problems , examples were the focal point of work on tangents . The standard results on the of tangent lines for polynomial graphs were obtained by Fermat in the monomial case and in complete generality by Hudde . Applications of to repeated roots of polynomials were also discovered at this time . Tangent lines to the cycloid were determined independently by Descartes , Fermat and Roberval .

Fermat also studied maximization and minimization problems using the approach he developed for tangent lines . We note that he was interested in several different types of minimization problems in mathematics and physics , including the determination of a point inside a triangle such that the sum of the to all three sides is minimized , and more significantly his Least Time Principle in optics , which states that a beam of light will take the path from one point to another that takes the shortest amount of time and yields the standard physical laws governing refraction and reflection . Further discussion of such basic minimization problems is contained in the following book :

S. Hildebrandt and A. Tromba. The Parsimonious Universe . Springer – Verlag, New York , 1996 . ISBN: 0 – 387 – 97991 – 3.

A result of Kepler ’s on maximization (the Wine Barrel Problem ) can also be mentioned at this point . He observed that a wine merchant had figured the amount of wine in a barrel by inserting a rod into the barrel diagonally through a small hole in the top . When the rod was removed , the length L of the rod which was wet determined the price for the barrel . Kepler was concerned about the uniformity of pricing by this method and decided to analyze its accuracy ; he correctly realized that a taller , narrower barrel might yield the same measurement as a shorter , wider one , resulting in the same wine price , even though its volume would be considerably smaller (see the figure below).

(Source: http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/apps/maxmin.html )

In his approach to determine the volume , Kepler approximated the barrel by a cylinder with base r of the base and height h; the conditions of the problem imply the 2 2 2 equation r + h = L . He then looked for the values of r and h giving the largest volume V when L (which determines the cost) is held constant . Using differential 2 2 calculus one can show that the relation between L and h must be 3 L = h , which is the answer that Kepler found using less refined methods. He also observed that the shapes of the wine barrels were close to this optimal value — in fact , so close that he could not believe this was a coincidence . Of course , the manufacturing processes then were less uniform than they are now , so is was unlikely that all barrels satisfied this mathematical specification precisely , but Kepler further noted that if a barrel deviated slightly from the optimal ratio this would have little effect on the volume because the volume function changes very slowly near its maximum .

Other contributions of Kepler faasdfdfsa

The emergence of calculus

Since many of the basic facts in calculus were known before the work of Newton and Leibniz , it is natural to ask why they are given credit for inventing the subject. Others came very close to doing so ; in particular , Barrow understood that the process for finding tangents (differentiation in modern terminology) was inverse to the process for finding areas (integration in modern terminology) . In these notes we shall focus on the decisive advances that make the work of Newton and Leibniz stand out from the important , high quality results due to many of their excellent contemporaries .

1. Before Newton and Leibniz techniques for differentiating and integrating specific examples had been developed , but they were the first to set general notation and define general “algorithmic processes ” for each construction . Earlier workers were not able to derive useful and general problem – solving methods.

2. Newton and Leibniz recognized the usefulness of differentiation and integration as general processes , not just as ad hoc methods to solve measurement and tangent problems in important special cases . Apparently no one (at least in Europe) had previously recognized the usefulness of calculus as a general mathematical tool .

3. With the exception of Barrow , the inverse relationship between differentiation and integration had not been clearly recognized in earlier work , and Newton and Leibniz were the first to formulate it explicitly and establish it in a logically convincing manner .

4. Both stated the main ideas and results of calculus algebraically , so that the subject was no longer an offshoot of classical Greek geometry but significantly broader in scope and poised to make fundamental contributions to many areas of knowledge .

Sections 8.3 and 8.4 of Burton contain a great deal of detail about the scientific and philosophical contributions of Newton (see pages 365 – 381) and Leibniz (see pages 383 – 402) . In particular , the bitter and heated dispute about credit for discovering calculus is described there . We shall only summarize the points that are now generally accepted : The discoveries of Newton and Leibniz were essentially independent , and although Newton was the first to develop the subject , Leibniz published his version first . We should add that the discoveries by Newton and Leibniz took place around 1665 and 1673 respectively . Leibniz ’ work was published in 1684 while Newton’s was published 1736 , nearly a decade after his death . Rather than dwell on the dispute over priorities , we shall discuss a few substantive similarities and differences between the work of Newton and Leibniz .

Many of the similarities were already mentioned in the reasons why Newton and Leibniz are given credit for creating calculus . One additional similarity is that each used both differentiation and integration to solve difficult and previously unsolved problems . Both also proved many of the same basic results ; e.g ., the standard rules for differentiating functions , the Fundamental Theorem of Calculus, and the basic formal integration techniques which appear in calculus textbooks . On the other hand , Newton and Leibniz clearly had different priorities and these can be seen in some differences between their approaches and conclusions.

a 1. The standard expansion for (1 + x) , where a is an

arbitrary real number and | x | < 1, is solely due to Newton .

2. Newton used the words and to denote the and integral, and he denoted derivatives by placing dots over variables . Leibniz ultimately adopted the dx notation and the integral sign that have been standard for centuries .

3. Newton was primarily interested in the uses of calculus to study problems involving motion , while Leibniz ’ work and interests involved finding extrema and solving differential equations .

4. Newton discovered the rules and processes of calculus by a study of velocity and distance , while Leibniz did so via algebraic sums and differences .

5. Newton used infinitesimals as a computational means, while Leibniz used them directly .

6. Newton ’s priority was differentiation while Leibniz ’ was integration .

7. Newton stressed the use of infinite series to express functions , while Leibniz preferred solutions that could be written in finite terms .

8. Leibniz gave more general rules and more convenient notation .

The other important writings of Newton and Leibniz reflect some of the differences mentioned above . Leibniz wrote many lengthy and highly influential works on philosophy , while Newton wrote several important books on the sciences . The latter include his best known work , Philosophiæ naturalis principia mathematica (The Mathematical Principles of Natural Philosophy) , which remains of the most important books in the sciences ever written . In this book he developed the laws of motion using calculus and used them to derive Kepler ’s laws of planetary motion . An extremely brief but informative summary of Principia is available at the following online site :

http://www.answers.com/topic/newton-s-principia

We have already mentioned some common aspects of Newton ’s and Leibniz ’ legacies with respect to calculus , and we shall conclude this discussion by mentioning some noteworthy differences :

1. Newton ’s applications of calculus ultimately determined the direction of subsequent work in mathematics and physics .

2. Leibniz ’ formulation of calculus ultimately determined how the mathematical aspects of this work were formulated (however , Newton ’s dot notation for derivatives is still used sometimes in physics to denote derivatives with respect to time) .

Other contributions of Newton and Leibniz fsdaasdf